A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neither the odd numbers nor the prime numbers tile the plane. We show that N can tile many, even infinitely many planes.

Let \( s, t \) be any numbers in \( \{0,1\} \) and let \( \pi = \pi_1 \pi_2 \cdots \pi_m \) be any word. We say that \( i \in [m-1] \) is an \( (s,t) \)-parity-rise if \( \pi_i \equiv s \pmod{2} \), \( \pi_{i+1} \equiv t \pmod{2} \), and \( \pi_i < \pi_{i+1} \). We denote the number of occurrences of \( (s,t) \)-parity-rises in \( \pi \) by \( \text{rise}_{s,t}(\pi) \). Also, we denote the total sizes of the \( (s,t) \)-parity-rises in \( \pi \) by \( \text{size}_{s,t}(\pi) \), that is, \( \text{size}_{s,t}(\pi) = \sum_{\pi_i < \pi_{i+1}} (\pi_{i+1} – \pi_i). \) A composition \( \pi = \pi_1 \pi_2 \cdots \pi_m \) of a positive integer \( n \) is an ordered collection of one or more positive integers whose sum is \( n \). The number of summands, namely \( m \), is called the number of parts of \( \pi \). In this paper, by using tools of linear algebra, we found the generating function that counts the number of all compositions of \( n \) with \( m \) parts according to the statistics \( \text{rise}_{s,t} \) and \( \text{size}_{s,t} \), for all \( s, t \).

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

We prove that if \( A \subset \mathbb{Z}_q \setminus \{0\} \), \( A \neq \langle p \rangle \), \( q = p^\ell \), \( \ell \geq 2 \) with \( |A| > C \sqrt[3]{\sqrt{\ell}^2 q^{(1-\frac{1}{4\ell})}} \), then
\[
|P(A) \cdot P(A)| \geq C’ q^3
\]
where
\[
P(A) = \left\{ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \in SL_2(\mathbb{Z}_q) : a_{11} \in A \cap \mathbb{Z}_q^\times, a_{12}, a_{21} \in A \right\}.
\]

The proof relies on a result in \([4]\) previously established by D. Covert, A. Iosevich, and J. Pakianathan, which implies that if \( |A| \) is much larger than \( \sqrt{\ell} q^{(1-\frac{1}{4\ell})} \), then
\[
|\{(a_{11}, a_{12}, a_{21}, a_{22}) \in A \times A \times A \times A : a_{11} a_{22} + a_{12} a_{21} = t\}| = |A|^4 q^{-1} + \mathcal{R}(t)
\]
where \( |\mathcal{R}(t)| \leq \ell |A|^2 q^{(1-\frac{1}{2\ell})} \).

By extending former results of Ehrhart, it was shown by Peter McMullen that the number of lattice points in the Minkowski-sum of dilated rational polytopes is a quasipolynomial function in the dilation factors. Here we take a closer look at the coefficients of these quasi-polynomials and show that they are piecewise polynomials themselves and that they are related to each other by a simple differential equation. As a corollary, we obtain a refinement of former results on lattice points in vector dilated polytopes

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Eulerian numbers \( A_{n,k} \) and Cauchy’s integral formula, asymptotic results in non-central region. In the region \( k = n – n^\alpha \), \( 1 > \alpha > 1/2 \), we analyze the dependence of \( A_{n,k} \) on \(\alpha\). This paper fits within the framework of Analytic Combinatorics.